By the late 1980s and early 1990s there was interest in attempting
to measure the topology of Large Scale Structure from observational
data and various models [1]}, [2]}, [3]}, [4]}, [5]}.
This was done using the genus statistic which is related to the fourth
Minkowski functional [6]}. These kinds of measures should
give an idea of the topological connectedness of a systems of points after they
have been smoothed by some kind of filter.
In the end this method allowed one to distinguish among different galaxy
distributions by obtaining the genus, using isodensity surfaces at different
density levels. These clearly require some kind of smoothing,
but the choice of levels at which to apply smoothing is not obvious.
This is important because
over-smoothing tends to create a positive genus, while under-smoothing
creates a negative one. Nonetheless these problems have not stopped groups from
applying these techniques to the largest available redshift surveys
of galaxies available at the moment, such as QDOT, CfA2, PSCz, 2dFGRS, and the
SDSS [7]}, [8]}, [9]}, [10]}, [11]}.
[12]} used Minkowski Functionals combined with percolation analysis to
compare the supercluster-void network in three cosmological models and that of
the present epoch.
Some of the latest studies [11]}, [14]} seem to confirm a sponge
like topology, and is consistent with the Gaussian random phase
initial conditions expected from inflation.
Recent work [15]}, [16]} has attempted to calculate Minkowski
Functionals using Delaunay Tessellation to calculate the isodensity
surfaces to try and get around the smoothing problem mentioned above.
